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arXiv · 1910.07305

Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

Yasuaki Kobayashi·Kensuke Kojima·Norihide Matsubara·Taiga Sone·Akihiro Yamamoto
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Abstract

The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph $G$ whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of $G$ having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in $O(n^3)$ time for trees and $O(n + m)$ time for split graphs and properly colored bipartite graphs, where $n$ is the number of vertices and $m$ is the number of edges. In this paper, we show that BCS can be solved in $O(n^2)$ time for trees and $O(n^3)$ time for interval graphs. The former result can be extended to bounded treewidth graphs. We also consider a weighted version of BCS (WBCS). We prove that this variant is weakly NP-hard even on star graphs and strongly NP-hard even on split graphs and properly colored bipartite graphs, whereas the unweighted counterpart is tractable on those graph classes. Finally, we consider an exact exponential-time algorithm for general graphs. We show that BCS can be solved in $2^{n/2}n^{O(1)}$ time. This algorithm is based on a variant of Dreyfus-Wagner algorithm for the Steiner tree problem.

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Yasuaki Kobayashi, Kensuke Kojima, Norihide Matsubara, Taiga Sone, Akihiro Yamamoto. 2019-10-16. Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem. https://arxiv.org/abs/1910.07305

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